Idea

An infinitesimal quantity is supposed to be a quantity that is infinitely small in size, yet not necessarily perfectly small (zero). An infinitesimal space is supposed to be a space whose extension is infinitely small, yet not necessarily perfectly small (pointline).

Infinitesimal objects have been conceived and used in one way or other for a long time, notably in algebraic geometry, where Grothendieck emphasized the now familiar role of formal duals (affine schemes) of commutative rings RR with nilpotent ideals J⊂RJ\subset R as infinitesimal thickenings of the formal dual of the quotient ring R/JR/J.

Formalization in synthetic differential geometry

A proposal for formalizing the abstract nonsense behind the notion of the infinitesimal such that these algebraic constructions become models for more general axioms was given by William Lawvere in his 1967 lecture (see the references below).

Lawvere observed that a simple yet powerful characterization of the notion of infinitesimal spaceDD is that DD is an object in a topos𝒯\mathcal{T} of spaces such that the inner homfunctor(−)D:𝒯→𝒯(-)^D : \mathcal{T} \to \mathcal{T} has a right adjoint.

If the topos in question furthermore is equipped with a line objectRR that plays the role of the real lineℝ\mathbb{R} then a sensible notion of infinitesimal quantities in RR is obtained when all morphisms D→RD \to R from infinitesimal spaces DD are necessarily linear maps. This is now known as the Kock-Lawvere axiom on lined toposes (𝒯,R)(\mathcal{T}, R). When it is satisfied, (𝒯,R)(\mathcal{T}, R) is called a smooth topos. The study of these is known as synthetic differential geometry.

The notion of infinitesimal object and infinitesimal space then makes sense in any smooth topos, and may be reasoned about generally for all smooth toposes. In any concrete model for the axioms there will accordingly be concrete realizations of these infinitesimal objects.

Comparison to infinitesimals in nonstandard analysis

Another notion of infinitesimals has arisen in the context of nonstandard analysis. The infinitesimal quantities considered there differ from the general ones in synthetic differential geometry in that they are all invertible (their inverses being “infinitely large”). Nevertheless, one can construct models of synthetic differential geometry which, in addition to nilpotent infinitesimals, contain invertible infinitesimals; see for instance MSIA, chapters VI and VII. Such invertible infinitesimals can be applied in some of the same ways as the infinitesimals of nonstandard analysis.

However, as pointed out in MSIA (intro. to Chapter VII), “there are some obvious differences.” The primary tool used in nonstandard analysis is a completely general transfer principle?, saying that any statement in the ordinary world is also true in the nonstandard world. In particular, this implies that the infinitesimal and infinitely large quantities in nonstandard analysis obey all the same rules of arithmetic and analysis as do the standard ones. By contrast, a limited sort of transfer principle relating a pair of specific models for SDG is proven in MSIA, but it applies only to statements of a certain logical form. Moreover, the arithmetic of invertible infinitesimals in SDG has some unfamiliar aspects: for instance, mathematical induction is only valid for statements of a certain logical form, and the axiom of finite choice fails.

The construction of models for nonstandard analysis does, however, have a topos-theoretic description, using filterpower?s.

as discussed in detail at sieve and sheaf. This says effectively that every point of XX is element of at least one of the covering spaces UiU_i and that one obtains XX by identifying the points in the covering spaces that correspond to the same one in XX.

Now let Σ\Sigma be any other space. We may assume here that the internal hom[Σ,−]:T→T[\Sigma,-] : T \to T at least preserves coproducts, so that applying this functor to the above diagram yields

Now notice how this will in general fail to still be a coequalizer: if it were, for one the morphism (∐i[Σ,Ui])→[Σ,X] (\coprod_i [\Sigma,U_i])
\to [\Sigma,X] would have to be an epimorphism. But this can’t be in general, because it would mean that every map Σ→X\Sigma \to X factors through one of the covering spaces. The problem here is that in general the image of Σ→X\Sigma \to X may be larger than any of the UiU_i.

This is maybe most familiar in the context of loop spaces (for Σ\Sigma the circle): the loop space of a cover of XX is not in general a cover of the loop space.

But suppose that Σ\Sigma were infinitesimal. One thing that should mean is that there is no other space that is “effectively smaller” in some useful sense. For Σ\Sigma infinitesimal, we do expect that every map Σ→X\Sigma \to X can always be factored through at least one of the UiU_i: because Σ\Sigma is so small, the image of a map out of it can never be too large.

So only if Σ\Sigma qualifies as having infinitesimal extension can the functor [Σ,−][\Sigma,-] be expected to preserve colimits.

Formal infinitesimal space

Definition (formal infinitesimal space)

An object Δ\Delta in a smooth topos(𝒯,R)(\mathcal{T}, R) is called a formally infinitesimal object if it is the algebra-spectrum of (what in the sdg-literature is usually called) a -RR-Weil algebra in 𝒯\mathcal{T}

SpecR(W):=RAlg𝒯(W,R)⊂RWSpec_R(W) := R Alg_{\mathcal{T}}(W,R) \subset R^W is the subobject of the internal hom of morphisms that respect the RR-algebra structure on WW and RR.

All the spaces that are described as collection of degree nn infinitesimal neighbours are of this form. Infinitesimal spaces not of this form are germ-spaces (see the examples below). These violate the finite-dimensionality assumption on JJ.

Examples

Infinitesimal intervals

There are several different objects that one may think of as an infinitesimal interval.

The smallest of them is often denoted DD and sometimes called the disembodied tangent vector or the walking tangent vector .

It is such that a morphism D→XD \to X into a manifoldXX is the same as a choice of point x∈Xx \in X and of a tangent vectorv∈TxXv \in T_x X. Equivalently, it is such that restricting a smooth function f:ℝ→ℝf : \mathbb{R} \to \mathbb{R} along the inclusion D↪ℝD \hookrightarrow \mathbb{R} produces the first-order jet defined by ff at the point 0↪D→ℝ0 \hookrightarrow D \to \mathbb{R}.

Accordingly, for each k∈ℕk \in \mathbb{N} there is a “slightly bigger” infinitesimal interval often denoted DkD_k, which is such that restricting a smooth function f:ℝ→ℝf : \mathbb{R} \to \mathbb{R} along Dk→ℝD_k \to \mathbb{R} produces the order-kkjet represented by this function at the given point.

Still infinitesimal but bigger than all these is the object Λ0:=∩0∈U⊂ℝU\Lambda_0 := \cap_{0 \in U \subset \mathbb{R}} U of intersections of all neighbourhods of the origin of ℝ\mathbb{R}. This is such that the restriction of a map f:ℝ→ℝf : \mathbb{R} \to \mathbb{R} along Λ0↪ℝ\Lambda_0 \hookrightarrow \mathbb{R} produces the germ of ff at 00.

The standard infinitesimal interval

Models

The classical example of a realization of an infinitesimal object is in terms of what is (traditionally but undescriptively) called the ring of dual numbers. For that we place ourselves in some context in which spaces are characterized dually in terms of the quantities on them, i.e. in terms of their would-be function algebras.

For some real number t∈ℝt \in \mathbb{R}, functions on the closed interval[−t,t]⊂ℝ[-t,t] \subset \mathbb{R} of length 2t2 t may be thought of as represented by functions on the whole real line ℝ\mathbb{R}, where two representatives represent the same function on the interval if they differ by a function that vanishes on the interval.

Precisely:

Lemma

The (generalized smooth) algebra of smooth functions C∞([−t,t])C^\infty([-t,t]) on [−t,t][-t,t] is isomorphic to the quotient of the algebra of smooth functions C∞(ℝ)C^\infty(\mathbb{R}) on all of ℝ\mathbb{R} by the functions that vanish on [−t,t][-t,t]

Proof

This is a corollary of the smooth version of the Tietze extension theorem, which says that for U⊂ℝnU \subset \mathbb{R}^n a closed subset, every smooth function on UU extends to a smooth function on all of ℝn\mathbb{R}^n.

As we think of the length of the interval shrinking to an infinitesimal value, the notion of derivative of functions is such that we want to say that the statement “a function vanishes on the infinitesimal interval” is equivalent to “a function vanishes at the origin and its first derivative there vanishes, too”. This in turn is usually equivalent (in a smooth context) to “a function is a square of a function that vanishes at the origin”.

Accordingly, in a context where one considers polynomial functions over the ground fieldkk, the infinitesimal interval is given by the space – usually called DD – that is dual to the ringk[ϵ]:=k[Z]/Z2k[\epsilon] := k[Z]/Z^2 which is the quotient of the polynomial ring in one variable ZZ modulo the polynomial Z2Z^2. This is often called the ring of dual numbers (where the term ‘dual’ historically refers to its being 22-dimensional). In terms of generators and relations this is the ring generated by a single element ϵ\epsilon subject to the relation that ϵ2=0\epsilon^2 = 0.

Definition (the 11-dimensional infinitesimal space)

In the context of generalized smooth algebra, the 11-dimensional infinitesimal space is the space DD whose function algebra is the quotient

C∞(D):=C∞(ℝ)/{x2}
C^\infty(D) := C^\infty(\mathbb{R})/\{x^2\}

of all functions on the real line, modulo those that are a product with the function x↦x2x \mapsto x^2.

This does reproduce the above ring of dual numbers due to the Hadamard lemma, which says that for g∈C∞(ℝ)g \in C^\infty(\mathbb{R}) a smooth function, there exists a smooth function h∈C∞(ℝ)h \in C^\infty(\mathbb{R}) such that for all x∈ℝx \in \mathbb{R} we have g(x)=g(0)+xg′(x)+x2h(x)g(x) = g(0) + x g'(x) + x^2 h(x). So modulo x2x^2, every smooth function is in fact a polynomial function.

In this dual generators-and-relations description, the infinitesimal interval is very familiar in many mathematically less sophisticated contexts. It prevails for instance in the basic physics textbook treatment since Isaac Newton up to this day. Sophus Lie is famously quoted as having said that he found many of his famous insights by such “synthetic reasoning” and only a lack of proper formalization prevented him from writing them up in this way instead of in the more wide-spread way of differential calculus.

The kk-dimensional infinitesimal disk

Example

Remark

Since in particular xj2=0x_j^2 = 0 for all elements of the infinitesimal nn-disk, we have an inclusion

D(n)⊂Dn
D(n) \subset D^n

which is proper if n>1n \gt 1. For n=1n = 1 we have D(1)=DD(1) = D.

While D(n)D(n) is closed under multiplication by elements of RR, it is not in general closed under addition of its elements. For instance for d1,d2∈D(1)=Dd_1,d_2 \in D(1) = D we have that d1+d2d_1 + d_2 (the operation being in RR) is still in DD precisely if (d1,d2)(d_1,d_2) is in D(2)D(2).

The infinitesimal neighbourhood

For x∈Xx \in X a point in a manifold, the infinitesimal neighbourhoodUpU_p is the intersection of all open neighbourhoods of xx. This is such that the restriction of a function f:X→ℝf : X \to \mathbb{R} along the inclusion Up→XU_p \to X is precisely the germ of the function ff.

All of the infinitesimal spaces above are contained in the corresponding infinitesimal neighbourhood. So this is the “largest” of the infinitesimal spaces discussed here.

we write D˜(k,n)\tilde D(k,n) for the space of all infinitesimal kk-simplices in RnR^n. More precisely, this is defined as the formal dual of the algebra C∞(D˜(k,n))C^\infty(\tilde D(k,n)) defined as follows.

Functions on spaces of infinitesimal kk-simplices turn out to be degree kk-differential forms. This provides a “synthetic” way of precisely thinking of wedge produts dx∧dyd x \wedge dy etc as products of infinitesimals. As the following computations do show, the skew-commutativity of the wedge product is an inherent consequence of the nature of infinitesimals.

A general element ff of this algebra we think of as a function on a certain infinitesimal neightbourhood of the origin of Rk⋅nR^{k \cdot n}, interpreted as the space of infinitesimal kk-simplices in RnR^n based at 0.

for f∈ℝf \in \mathbb{R} and (a,b,ω,λ∈(ℝn)*)1≤i≤n(a, b, \omega, \lambda \in (\mathbb{R}^n)^*)_{1 \leq i \leq n} a collection of ordinary covectors and with “⋅\cdot” denoting the evident contraction, and where in the last step we used the above relations.

It is noteworthy here that the coefficient of the term which is multilinear in each of the ϵi\epsilon_i is the wedge product of two covectors ω\omega and λ\lambda: we may naturally identify the subspace of C∞(D˜(2,2))C^\infty(\tilde D(2,2)) on those elements that vanish if either ϵ1\epsilon_1 or ϵ2\epsilon_2 are set to 0 as the space ∧2T0*ℝ2\wedge^2 T_0^* \mathbb{R}^2 of 2-forms at the origin of ℝ2\mathbb{R}^2.

Of course for this identification to be more than a coincidence we need that this is the beginning of a pattern that holds more generally. But this is indeed the case.

Properties

Let EE be the set of square submatrices of the k×nk \times n-matrix (ϵij)(\epsilon_i^j). As a set this is isomorphic to the set of pairs of subsets of the same size of {1,⋯,k}\{1, \cdots, k\} and {1,⋯,n}\{1, \cdots , n\}, respectively. For instance the square submatrix labeled by {2,3,4}\{2,3,4\} and {1,4,5}\{1,4,5\} is

for the corresponding determinant, given as a product of generators in C∞(D˜(k,n))C^\infty(\tilde D(k,n)). Here the sum runs over all permutations σ\sigma of {1,⋯,r}\{1, \cdots, r\} and sgn(σ)∈{+1,−1}⊂ℝsgn(\sigma) \in \{+1, -1\} \subset \mathbb{R} is the signature of the permutation σ\sigma.

Proposition

The elements f∈C∞(D˜(k,n))f \in C^\infty(\tilde D(k,n)) are precisely of the form

f=∑e∈Efedet(e)
f = \sum_{e \in E} f_e \; det(e)

for unique{fe∈ℝ|e∈E}\{f_e \in \mathbb{R} | e \in E\}. In other words, the map of vector spaces

By the relations in C∞(D˜(k,n))C^\infty(\tilde D(k,n)), this is non-vanishing precisely if none of the ii-indices repeats and none of the jj-indices repeats. Furthermore by the relations, for any permutation σ\sigma of rr elements, this is equal to

Remark

effectively this proposition appears as the “Kock-Lawvere axiom scheme for D˜(k,n)\tilde D(k,n)” when D˜(k,n)\tilde D(k,n) is regarded as an object of a suitable smooth topos. It is useful to record this simple but very crucial observation of Anders Kock here in the category AlgℝopAlg_{\mathbb{R}}^{op} or in the category C∞AlgopC^\infty Alg^{op} of smooth loci, as we do here, where it is just a simple observation. The point of the Kock-Lawvere axiom scheme is effectively to ensure that the properties of C∞(D˜(k,n))∈C∞AlgopC^\infty(\tilde D(k,n)) \in C^\infty Alg^{op} are preserved under Yoneda embedding into a corresponding sheaf topos. But it has been observed that it serves to clarify what is going on in parts of Ander Kock’s book by separating the combinatorial and algebraic arguments from their internalization into suitable smooth toposes.

Let C∞(D˜(k,n))topC^\infty(\tilde D(k,n))_{top} be the sub-vector space of the underlying vector space of C∞(D˜(k,n))C^\infty(\tilde D(k,n)) on those elements that vanish if the collection of generators ϵi=(ϵi1,ϵi2,⋯,ϵin)\epsilon_i = (\epsilon_i^1 , \epsilon_i^2, \cdots, \epsilon_i^n) is set to 0, for all ii. This are those elements that are linear combinations of the form ∑etop∈Etopdet(etop)fetop\sum_{e_{top} \in E_{top}} det(e_{top}) f_{e_{top}}, for etope_{top} ranging over the maximal square submatrices of (ϵij)(\epsilon_i^j).

So inside the space of functions on infinitesimal simplices, we find the differential forms. The next crucial observation now is that there is a natural reason , from the nPOV, to restrict to C∞(D˜(k,n))top⊂C∞(D˜(k,n))C^\infty(\tilde D(k,n))_{top} \subset C^\infty(\tilde D(k,n)).

The tangent Lie algebroid and differential forms

The collection of the spaces Rn×D˜(k,n)R^n \times \tilde D(k,n) for all k∈ℕk \in \mathbb{N} naturally forms a simplicialsmooth locus(ℝn)(Δinf•)(\mathbb{R}^n)^{(\Delta^\bullet_{inf})}, which represents the infinitesimal path ∞-groupoid? of ℝn\mathbb{R}^n, equivalently the tangent Lie algebroid of ℝn\mathbb{R}^n.

Dually this is a smoothcosimplicial algebra. Under the normalized cochain complex functor of the dual Dold-Kan correspondence this identifies with a dg-algebra. The fact that this is the normalized cochain complex algebra means that it consists in degree kk only of a subspace of the space that the cosimplicial algebra has in degree kk. This subspace is precisely that of differential kk-forms.

This we now describe in detail. All the arguments involved are still (with slightly different parameterization, possibly) due to Anders Kock, the only new thing here being the observation that the restriction to the joint kernel of the degeneracy maps exhibts the Dold-Kan map, and that this way using the simplicial picture everything acquires a nice nPOV interpretation as being about the infinitesimal path ∞-groupoid? of ℝn\mathbb{R}^n, regarded either as an infinitesimal Lie ∞-groupoid or as a ∞-Lie algebroid.

where tildeetilde e is obtained from ee by replacing each ϵij\epsilon_{i}^j it contains with ϵi+1j\epsilon_{i+1}^j.

The way to think of how the face and degeracy maps here work is to imagine that a collection of elements (v1,⋯,vk)∈D˜(k,n)(v_1, \cdots, v_k) \in \tilde D(k,n) spans an infinitesimal kk-parallelepiped, and that inside that the face and degeneracy maps slice out a kk-simplex. The proof that this is indeed a (co)simplicial object is entirely analogous to the discussion of the simplicial object of finite simplices at interval object.

For instance for k=3k = 3 we have six 3-simplices sitting inside each 3-cube

and the face maps identify one of these:

.

Now oberserve that under the dual Dold-Kan correspondence the normalized cochain complex of this cosimplicial algebra is, up to isomorphism, the complex that in degree kk has the joint kernel of the co-degeneracy maps. But by the above remarks, this joint kernel is precisely